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33.14 Change of fields and the Jacobson property

A scheme locally of finite type over a field has plenty of closed points, namely it is Jacobson. Moreover, the residue fields are finite extensions of the ground field.

Lemma 33.14.1. Let $X$ be a scheme which is locally of finite type over $k$. Then

  1. for any closed point $x \in X$ the extension $\kappa (x)/k$ is algebraic, and

  2. $X$ is a Jacobson scheme (Properties, Definition 28.6.1).

Proof. A scheme is Jacobson if and only if it has an affine open covering by Jacobson schemes, see Properties, Lemma 28.6.3. The property on residue fields at closed points is also local on $X$. Hence we may assume that $X$ is affine. In this case the result is a consequence of the Hilbert Nullstellensatz, see Algebra, Theorem 10.34.1. It also follows from a combination of Morphisms, Lemmas 29.16.8, 29.16.9, and 29.16.10. $\square$

It turns out that if $X$ is not locally of finite type, then we can achieve the same result after making a suitably large base field extension.

Lemma 33.14.2. Let $X$ be a scheme over a field $k$. For any field extension $K/k$ whose cardinality is large enough we have

  1. for any closed point $x \in X_ K$ the extension $\kappa (x)/K$ is algebraic, and

  2. $X_ K$ is a Jacobson scheme (Properties, Definition 28.6.1).

Proof. Choose an affine open covering $X = \bigcup U_ i$. By Algebra, Lemma 10.35.12 and Properties, Lemma 28.6.2 there exist cardinals $\kappa _ i$ such that $U_{i, K}$ has the desired properties over $K$ if $\# (K) \geq \kappa _ i$. Set $\kappa = \max \{ \kappa _ i\} $. Then if the cardinality of $K$ is larger than $\kappa $ we see that each $U_{i, K}$ satisfies the conclusions of the lemma. Hence $X_ K$ is Jacobson by Properties, Lemma 28.6.3. The statement on residue fields at closed points of $X_ K$ follows from the corresponding statements for residue fields of closed points of the $U_{i, K}$. $\square$

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